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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution</dfn> The characteristic equation is</p>
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\begin{equation*}
r^2-6r+9=0~\to~(r-3)^2=0~\to~r_1=r_2=3.
\end{equation*}
</div>
<p class="continuation">One solution is</p>
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\begin{equation*}
y_1=e^{3 x}.
\end{equation*}
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<p class="continuation">The second solution is</p>
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\begin{equation*}
y_2=x y_1=x e^{3x}.
\end{equation*}
</div>
<p class="continuation">The general solution is</p>
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\begin{equation*}
y=C_1 e^{3x}+C_2 x e^{3x}.
\end{equation*}
</div>
<p class="continuation">The initial conditions yield</p>
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\begin{equation*}
\begin{aligned}
&amp;y(0)=0:\quad 0=C_1 ~\to~y=C_2 x e^{3x}\\
&amp;y^{\prime}(0)=2 ~\to~2=C_2.
\end{aligned}
\end{equation*}
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<p class="continuation">The solution is</p>
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\begin{equation*}
y=2 x e^{3x}.
\end{equation*}
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<span class="incontext"><a href="sec3_5.html#p-108" class="internal">in-context</a></span>
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